The term rush means a way for a new character to quickly finish all the quests in Diablo II and Diablo II: Lord of Destruction. It is usually done with the help of more experienced players in order to progress faster. This is called boosting or training (with the "leading" player dubbed loco and the new character(s) called cars, play of words: train and training).
In Diablo III, similar activities are sometimes performed: the new character (mostly one replacing a deceased Hardcore character) is simply following the leading comrade(s) at the maximum possible difficulty, gaining experience and items at a vastly increased rate. At high Torment difficulty, the process of leveling from 1 to 70 may take less than 1 minute with 3 supporting players.